A long ago my son Sergei went to the Streamline School Olympiad. Some of the problems were really nice and I asked the organizer, Oleg Kryzhanovsky, where he took the problems from. It seems that he himself supplied all the problems, many of which are his original creations. He told me that he can invent a math Olympiad problem on demand for any level of difficulty on any math topic. No wonder that he is the author of almost all math problems at the Ukraine Olympiad.

The following is a sample of his problems from the Streamline Olympiad. For my own convenience I have chosen problems without figures and equations. Note: I edited some of them.

1998 (8th – 9th grade). Find three numbers such that each of them is a square of the difference of the two others.

1999 (9th – 10th grade). The positive integers 30, 72, and N have a property that the product of any two of them is divisible by the third. What is the smallest possible value of N?

1999 (9th – 10th grade). You have 6 coins weighing 1, 2, 3, 4, 5 and 6 grams that look the same. The number (1, 2, 3, 4, 5, 6) on the top of each coin should correspond to its weight. How can you determine whether all the numbers are correct, using the balance scale only twice?

1999 (11th – 12th grade). In how many ways can the numbers 1, 2, 3, 4, 5, 6 be ordered such that no two consecutive terms have a sum that is divisible by 2 or 3.

2000 (6th – 7th grade). Let A be the least integer such that the sum of all its digits is equal to 2000. Find the left-most digit of A.

2000 (8th grade). You have six bags with coins that look the same. Each bag has an infinite number of coins and all coins in the same bag weigh the same amount. Coins in different bags weigh 1, 2, 3, 4, 5 and 6 grams exactly. There is a label (1, 2, 3, 4, 5, 6) attached to each bag that is supposed to correspond to the weight of the coins in that bag. You have only a balance scale. What is the least number of times do you need to weigh coins in order to confirm that the labels are correct?

Sean:

Sam:

About the 1998 8th/9th problem:
That one took me a while too. {0,0,0} and {0,1,1} both seem to work and if you restrict the possibilities to real numbers, I believe you can pretty easily prove those are the only solutions. I haven’t tried to figure out it any solutions exist for imaginary or complex numbers.

I came to leave a similar comment. I go straight for checking extremes, but students often do not have that instinct. For that reason, I believe the 1998 may be the hardest of the group. Hmmph. The 1998 is also accessible to younger students — who — sounds strange — might provide a solution more quickly.

Austin:

That 1998 problem confused me too. Actually, I think the first thing I tried was exactly the right approach, but the answer (same as Sam’s) was so surprising that I thought I must have made a mistake. The phrasing of the question is misleading — it might be better if it said, “Find all such triples with proof,” which would be reasonable for an olympiad. Here is my solution:

We know a=(b-c)^2 and b=(a-c)^2.
Subtract the second equation from the first, and factor to obtain
a-b = (a-b)(2c-a-b).
So either a=b, or a+b=2c.

The same conclusion holds if we permute a,b,c. But if all permutations of a+b=2c hold, then it’s easy to see that a=b=c. So without loss of generality, a=b must hold, and c=0. Also, a=(b-0)^2=a^2, so the only possible solutions are a=1 or a=0.

I am sorry I misunderstood Oleg. I thought that all the problems from the Streamline Olympiad were his, while in fact it wasn’t true for earlier years. It looks like my favorite problem (1999 coins problem), which is the most difficult on the list is his. Here are the corrections Oleg sent me:

1. All 1998-problems were NOT mine. I just came to USA 1 day before
Olympiad took place :-))). I think that all 1998 problems were taken (by
Rachel Kamisnky) from some other sources, so they were not original
ones. You can refer to the problem as “Streamline School Olympiad
Problem”.

2. Some of 1999 problems were suggested by Rachel Kaminsky (she took
them from some other sources), others were suggested by myself (and some
of them were indeed original ones)

The first 1999 problem you mentioned was suggested by Rachel, the second
one was created by myself, and the third one was suggested (either by
Rachel or by myself, I cannot tell now for sure), it was not an original
one.

3. The 2000 problem you mentioned was created by myself.

4. At the end of the first paragraph it should be “Kharkov (Ukraine)
Olympiad” instead of “Ukraine Olympiad”

5. I can provide a simple solution for 1998 problem in the blog (of
course, it would be better to say “real numbers” instead of just
“numbers”, but “other source” did not say that, and I did not have a
chance to review the problems before the Olympiad :-))) )

Konstantin Knop provided me with the following reference for the six-coins problem. This problem appeared at the last round of Moscow math Olympiad in 1991 (for 8th grade). The author of this problem is Sergey Tokarev:http://kvant.mirror1.mccme.ru/1991/09/p70.htm